条件单向简单随机游动与表示论
Conditioned one-way simple random walk and representation theory
摘要：
我们称单向简单随机游动为象限Z_+^n中增量属于标准基的随机游动。结合李代数和超代数的表示论，我们描述了这样一个随机游动在闭八分圆、半开八分圆或其它类型的半群中的规律。这些代数的组合表示论使我们能够描述一个广义的皮特曼变换，实现了条件的一组路径的步行。我们在这里追求由O'Connell和他的合著者[13，14，2]发起的方向，并在[12]中发展。我们的工作依赖于晶体碱基理论和Kashiwara及其合著者在[1]和最近在[5]中描述的插入方案。
英文摘要：
We call one-way simple random walk a random walk in the quadrant Z_+^n whose increments belong to the canonical base. In relation with representation theory of Lie algebras and superalgebras, we describe the law of such a random walk conditioned to stay in a closed octant, a semi-open octant or other types of semi-groups. The combinatorial representation theory of these algebras allows us to describe a generalized Pitman transformation which realizes the conditioning on the set of paths of the walk. We pursue here in a direction initiated by O'Connell and his coauthors [13,14,2], and also developed in [12]. Our work relies on crystal bases theory and insertion schemes on tableaux described by Kashiwara and his coauthors in [1] and, very recently, in [5].
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